Question

Prove that `2 - 3sqrt(5)` is an irrational number.

Answer 1

Let us assume that \[2 - 3\sqrt{5}\] is rational. Then, there exist positive co-primes a and b such that

`2-3sqrt5=a/b`

`3sqrt5=a/b-2`

`3sqrt5=(a/b-2)/3`

`sqrt5=(a-2b)/(3b)`

This contradicts the fact `sqrt5` is an irrational number.

Hence `2-3sqrt5` is irrational.